EMP: Effective Multidimensional Persistence for Graph Representation Learning

TL;DR

This paper introduces Effective Multidimensional Persistence (EMP), a novel topological data analysis framework that captures complex data features by analyzing multiple scale parameters simultaneously, improving graph classification performance.

Contribution

The paper presents EMP, a new multidimensional persistence framework that extends traditional persistent homology to multiple parameters, with theoretical guarantees and practical effectiveness in graph learning.

Findings

EMP outperforms existing methods on benchmark datasets.

EMP enhances the expressiveness of topological summaries.

The framework provides stability guarantees for multidimensional persistence.

Abstract

Topological data analysis (TDA) is gaining prominence across a wide spectrum of machine learning tasks that spans from manifold learning to graph classification. A pivotal technique within TDA is persistent homology (PH), which furnishes an exclusive topological imprint of data by tracing the evolution of latent structures as a scale parameter changes. Present PH tools are confined to analyzing data through a single filter parameter. However, many scenarios necessitate the consideration of multiple relevant parameters to attain finer insights into the data. We address this issue by introducing the Effective Multidimensional Persistence (EMP) framework. This framework empowers the exploration of data by simultaneously varying multiple scale parameters. The framework integrates descriptor functions into the analysis process, yielding a highly expressive data summary. It seamlessly integrates established single PH summaries into multidimensional counterparts like EMP Landscapes, Silhouettes, Images, and Surfaces. These summaries represent data's multidimensional aspects as matrices and arrays, aligning effectively with diverse ML models. We provide theoretical guarantees and stability proofs for EMP summaries. We demonstrate EMP's utility in graph classification tasks, showing its effectiveness. Results reveal that EMP enhances various single PH descriptors, outperforming cutting-edge methods on multiple benchmark datasets.